Kenneth W. Chase

A Survey of Research in the Application of Tolerance Analysis to the Design of Mechanical Assemblies

Tolerance analysis is receiving renewed emphasis as industry recognizes that tolerance management is a key element in their programs for improving quality, reducing overall costs, and retaining market share. The specification of tolerances is being elevated from a menial task to a legitimate engineering design function. New engineering models and sophisticated analysis tools are being developed to assist design engineers in specifying tolerances on the basis of performance requirements and manufacturing considerations. This paper presents an overview of tolerance analysis applications to design with emphasis on recent research that is advancing the state of the art. Major topics covered are (1) new models for tolerance accumulation in mechanical assemblies, including the Motorola Six Sigma model; (2) algorithms for allocating the specified assembly tolerance among the components of an assembly; (3) the development of 2-D and 3-D tolerance analysis models; (4) methods which account for non-normal statistical distributions and nonlinear effects; and (5) several strategies for improving designs through the application of modern analytical tools.

Generalized 3-D tolerance analysis of mechanical assemblies with small kinematic adjustments

The direct linearization method (DLM) for tolerance analysis of 3-D mechanical assemblies is presented. Vector assembly models are used, based on 3-D vector loops which represent the dimensional chains that produce tolerance stackup in an assembly. Tolerance analysis procedures are formulated for both open and closed loop assembly models. The method generalizes assembly variation models to include small kinematic adjustments between mating parts. Open vector loops describe critical assembly features. Closed vector loops describe kinematic constraints for an assembly. They result in a set of algebraic equations which are implicit functions of the resultant assembly dimensions. A general linearization procedure is outlined, by which the variation of assembly parameters may be estimated explicitly by matrix algebra. Solutions to an over-determined system or a system having more equations than unknowns are included. A detailed example is presented to demonstrate the procedures of applying the DLM to a 3-D mechanical assembly.

Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies

Geometric feature variations are the result of variations in the shape, orientation or location of part features as defined in ANSI Y14.5M-1982 tolerance standard. When such feature variations occur on the mating surfaces between components of an assembly, they affect the variation of the completed assembly. The geometric feature variations accumulate statistically and propagate kinematically in a similar manner to the dimensional variations of the components in the assembly.The direct linearization method (DLM) for assembly tolerance analysis provides a means of estimating variations and assembly rejects, caused by the dimensional variations of the components in an assembly. So far no generalized approach has been developed to include all geometric feature variations in a computer-aided tolerance analysis system.This paper introduces a new, generalized approach for including all the geometric feature variations in the tolerance analysis of mechanical assemblies. It focuses on how to characterize geometri...

A Comprehensive System for Computer-Aided Tolerance Analysis of 2-D and 3-D Mechanical Assemblies

Tolerance analysis of assemblies promotes concurrent engineering by bringing engineering requirements and manufacturing capabilities together in a common model. By further integrating the engineering modeling and analysis with a CAD system, a practical tool for product and process development is created. It provides a quantitative design tool for predicting the effects of manufacturing variation on performance and cost in a computer-based design environment.

Direct Linearization Method Kinematic Variation Analysis

Velocity and acceleration analysis is an important tool for predicting the motion of mechanisms. The results, however, may be inaccurate when applied to manufactured products due to the process variations that occur in production. Small changes in mechanism dimensions can accumulate and propagate, causing a significant variation in the performance of the mechanism. A new application of statistical analysis is presented for predicting the effects of variation on mechanism kinematic performance. The new method is an extension of the direct linearization method developed for static assemblies. This method provides a solution that is a closed form. It may be applied to two-dimensional mechanisms to predict variation in velocity and acceleration due to dimensional variations. It is also shown how form, orientation, and position variations may be included in the analysis to analyze variations that occur within the joints. Only two assemblies are analyzed to characterize the distribution: The first determines the mean, and the second estimates the variance. The system is computationally efficient and well suited for design iteration.

Variation Analysis of Tooth Engagement and Loads in Involute Splines

Involute spline couplings are used to transmit torque from a shaft to a hub or other rotating component. In theory, all teeth of the spline share the load equally. In practice, due to manufacturing variations, the teeth are unequally loaded. A new model for tooth engagement, based on statistics, predicts that the teeth engage in a sequence, determined by the individual clearances. As the shaft load is applied, the tooth pair with the smallest clearance engages first, then deflects as the load increases, until the second pair engage. The two engaged pairs deflect together until the third pair engage, and so on, until the full load is reached. The statistical model predicts the average number of teeth which will engage for a specified load, plus or minus the expected variation. It also quantitatively predicts the load and stress in each engaged pair. Critical factors in the model are the stiffness and deflection of a single tooth pair and the characterization of the clearance. Detailed finite element analyses were conducted to verify the tooth deflections and engagement sequence. This model has led to a simple closed-form solution that has been implemented in a spreadsheet to allow designers to predict the load in spline teeth based upon the characteristics of the spline.

Variation Simulation of Fixtured Assembly Processes for Compliant Structures Using Piecewise-Linear Analysis

While variation analysis methods for compliant assemblies are becoming established, there is still much to be done to model the effects of multi-step, fixtured assembly processes statistically. A new method is introduced for statistically analyzing compliant part assembly processes using fixtures. This method yields both a mean and a variant solution, which can characterize an entire population of assemblies. The method, called Piecewise-Linear Elastic Analysis, or PLEA, is developed for predicting the residual stress, deformation and springback variation resulting from fixtured assembly processes. A comprehensive, step-by-step analysis map is presented for introducing dimensional and surface variations into a finite element model, simulating assembly operations, and calculating the error in the final assembly. PLEA is validated on a simple, laboratory assembly and a more complex, production assembly. Significant modeling issues are resolved as well as the comparison of the analytical to physical results.Copyright

Smart Assemblies for Robust Design: A Progress Report

Research conducted over the past decade has resulted in a suite of methods for robust design which can be applied during different design stages. These methods focus on reducing the sensitivity of the design to variation without removing its causes. In this research we are investigating an additional and very powerful means for achieving robustness that complements the other methods developed to date. We have dubbed this area “smart assemblies.” A smart assembly has features, not otherwise required by the function of the design, which allow the design to absorb or cancel out the effects of variation. In this paper we report our results to date. We discuss the close connection between smart assembly design and exactly constrained design. We present the beginnings of a smart feature classification system, a preliminary methodology for smart assembly design, and a case study.Copyright

Variation Analysis of Position, Velocity, and Acceleration of Two-Dimensional Mechanisms by the Direct Linearization Method

Velocity and acceleration analysis is an important tool for predicting the motion of mechanisms. The results, however, may be inaccurate when applied to manufactured products, due to the process variations which occur in production. Small changes in dimensions can accumulate and propagate in an assembly, which may cause significant variation in critical kinematic performance parameters. A new statistical analysis tool is presented for predicting the effects of variation on mechanism kinematic performance. It is based on the Direct Linearization Method developed for static assemblies. The solution is closed form, and may be applied to 2-D, open or closed, multi-loop mechanisms, employing common kinematic joints. It is also shown how form, orientation, and position variations may be included in the analysis to analyze variations that occur in kinematic joints. Closed form solutions eliminate the need of generating a large set of random assemblies, and analyzing them one-by one, to determine the expected range of critical variables. Only two assemblies are analyzed to characterize the entire population. The first determines the performance of the mean, or average assembly, and the second estimates the range of variation about the mean. The system is computationally efficient and well suited for design iteration.© 2009 ASME

Rack and Pinion Steering Linkage Synthesis Using an Adapted Freudenstein Approach

Rack and pinion steering systems have been synthesized and designed for many automotive applications. A steering system must be optimized for best steering characteristics to reduce tire wear and to assure safety and stability. Unfortunately, each design is slightly different because of differing car parameters and different space constraints. Designers need a tool that can quickly provide optimally synthesized steering systems. A precise and efficient method has been developed to assist designers in finding an optimum planar mechanism design for rack and pinion steering systems. It can be used to design central take-off or side take-off steering systems that are in a leading or trailing configuration. This method combines a modified Freudenstein equation with numerical optimization. Because of the combination of methods, an optimized solution may be found quickly. Thus, the tool is well adapted for preliminary designs and for design iteration.Copyright